CENTRAL TENDENCY: MEAN, MEDIAN, MODE

ASSIGNMENTS

•  Pollock, Essentials, chs. 2-3

•  Begin exploring Pollock, SPSS Companion, introduction and chs. 1-2

•  Sections in Solís 105

Why Central Tendency?

•  Summation of a variable

•  Possibilities of comparison: – Across time – Across categories (e.g. countries, classes) – But beware of limitations!

Outline: Measures of Central Tendency

Mean or “average” value:  Definition  Illustration  Special properties (for future reference)

Median or middle value:  Definition  Comparison with mean

Mode or most frequent value:  Definition  Applications

Computing the Arithmetic Mean

Finding the Arithmetic Mean

Properties of the Arithmetic Mean

•  Data: 3, 4, 5, 6, 7 •  Mean = 25/5 = 5

Differences from mean: Squared differences: 3 – 5 = -2 4 4 – 5 = -1 1 5 – 5 = 0 0 6 – 5 = +1 1 7 – 5 = +2 4

Sum (Σ) = 0 Sum (Σ) = 10

Illustration: Properties of the Arithmetic Mean

Treating Other Numbers (e.g., 3) as Mean:

Differences: Squared differences: 3 – 3 = 0 0 4 – 3 = +1 1 5 – 3 = +2 4 6 – 3 = +3 9 7 – 3 = +4 16

Sum (Σ) = 10 [not zero] Sum = 30

Sum of squared differences for 4 = 15 Sum of squared differences for 6 = 15 Sum of squared differences for 7 = 30

True Mean (5) has least sum2 = 10

• Definition: The most frequent value • Especially useful for categorical variables • Focus upon the distribution of values • Applicable to quantitative data through

shape of distribution • Illustrations: heart disease, alcohol consumption, GDP per capita

Interpreting the Mode

Cautionary Tales

• Mean is the most common measure of central tendency

• It is misleading for variables with skewed distributions (e.g., power, wealth, education, income)

• Mean, median, and mode converge with normal distributions, and diverge in cases of skewed distributions.